18 years ago someone took one out in my name at a previous address I no longer lived at, and ran up a few thousand pounds of debt before the company found me, that took quite a bit of sorting, and a good long interview with CID to explain that I did not, nor ever had at that time, actually had a credit card. For years there was a flag against my name (and also the wifes despite her not being directly involved) meaning I had to jump through hoops and give extra details just to open a bank account.
A few years ago I did actually get a credit card, nice shiny gold thing from Barclays, then one day I somehow managed to use it in Inverness which was actually me, but somehow I bought a few hundred quids worth of car parts in Spain about an hour later....which definately wasn't me.... then there was the day I filled up at a petrol station and the bloody card would not work, no reason why, just wouldn't go through, I felt a right pratt after making the trip home to get actual real money when I walked in the garage and got in a condecending voice in front of a queue of customers "oh, youve come to pay for your fuel have you now"
Chuffing thing got the scissors when I got back home, never again
Ok, you are protected and can sort such things out, but what a bloody faff.
As Terry says, a Visa debit card has reasonable protection, and PayPal is pretty good for online payments.
To be honest I've never had a problem buying anything from a business that can't be sorted through the correct procedures, or a word with the supplier in the case of it being local. I suppose a lot depends on where & who you buy off.